Local logarithmic stability of an inverse coefficient problem for a singular heat equation with an inverse-square potential

2021 ◽  
pp. 1-23
Author(s):  
Xue Qin ◽  
Shumin Li
Keyword(s):  
Heat Equation ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Logarithmic Stability ◽  
Inverse Coefficient

Simultaneously identifying the thermal conductivity and radiative coefficient in heat equation from Dirichlet and Neumann boundary measured outputs

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Thermal Conductivity ◽  
Heat Equation ◽  
Explicit Formula ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity k(x) and radiative coefficient q(x) in the 1D heat equation u_{t}=(k(x)u_{x})_{x}-q(x)u from the most available Dirichlet and Neumann boundary measured outputs. The Neumann-to-Dirichlet and Neumann-to-Neumann operators \Phi[k,q](t):=u(\ell,t;k,q), \Psi[k,q](t):=-k(0)u_{x}(0,t;k,q) are introduced, and main properties of these operators are derived. Then the Tikhonov functionalJ(k,q)=\tfrac{1}{2}\lVert\Phi[k,q]-\nu\rVert^{2}_{L^{2}(0,T)}+\tfrac{1}{2}\lVert\Psi[k,q]-\varphi\rVert^{2}_{L^{2}(0,T)}of two functions k(x) and q(x) is introduced, and an existence of a quasi-solution of the inverse coefficient problem is proved. An explicit formula for the Fréchet gradient of the Tikhonov functional is derived through the weak solutions of two appropriate adjoint problems.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output

2018 ◽  
Vol 26 (3) ◽  
pp. 349-368 ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Parabolic Equation ◽  
Lipschitz Continuity ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Iteration Algorithm ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional {J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}} corresponding to an inverse coefficient problem for the {1D} parabolic equation {u_{t}=(k(x)u_{x})_{x}} with the Neumann boundary conditions {-k(0)u_{x}(0,t)=g(t)} and {u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operator\Phi[k]:=u(x,t;k)\lvert_{x=0^{+}},\quad\Phi[\,\cdot\,]:\mathcal{K}\subset H^{1% }(0,l)\mapsto H^{1}(0,T),as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output {f(t):=u(0,t;k)} are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient. This lemma shows that an important advantage of gradient methods comes when dealing with the functionals of class {C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if {J\in C^{1,1}(\mathcal{K})} and {\{k^{(n)}\}\subset\mathcal{K}} is the sequence of iterations obtained by the Landweber iteration algorithm {k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for {\omega_{n}\in(0,2/L_{g})}, where {L_{g}>0} is the Lipschitz constant, the sequence {\{J(k^{(n)})\}} is monotonically decreasing and {\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

Sign in / Sign up

Export Citation Format

Share Document